Transformation Groups Seminar

Speaker: Kumar Sannidhya Shukla (Western)

"Complexity 0 torus action on manifolds (Part 3)"

Time: 10:30

Room: MC 108

Let T be an n-dimensional torus acting on a ‘nice’ 2n-manifold M effectively, with nonempty set of fixed points and suppose that all the isotropy groups are connected. If the action satisfies another hypothesis (equivariant formality) then the quotient space M/T has the structure of a homology cell complex and is in fact a homology disk. In this talk we will give first discuss some general facts about orbits of torus actions on manifolds and about locally standard actions. Then using the Atiyah-Bredon-Franz-Puppe sequence we give a detailed proof of the fact that under the assumptions stated above, M/T is a homology disk.

Transformation Groups Seminar

Speaker: Lewis Stanton (University of Southampton)

"Loop spaces of moment-angle complexes associated to flag complexes"

Time: 10:30

Room: Zoom

Polyhedral products are natural subspaces of the Cartesian product of spaces, which have a diverse range of applications across mathematics. One particular special case is the moment-angle complex. Work of various authors has identified families of simplicial complexes for which the loop space of their corresponding moment-angle complex is homotopy equivalent to a product of spheres and loops on spheres. In this talk, I will survey the current progress in this direction, and then expand the family of simplicial complexes for which such a decomposition is known - namely simplicial complexes which are the k-skeleton of a flag complex.

Analysis Seminar

Speaker: Rasul Shafikov (Western)

"Rational convexity and symplectic geometry"

Time: 14:30

Room: MC 108

In this talk I will define Liouville vector fields, Weinstein domains, contact type submanifolds and discuss their connection with Stein structures and rational convexity in $\mathbb C^n$.

Geometry and Topology

Speaker: Lukas Muller (Perimeter Institute)

"Quantum representations of handlebody groups"

Time: 15:30

Room: MC 107

Mapping class groups of surfaces and handlebody groups are fundamental objects in low-dimensional topology. Quantum algebra and mathematical physics provide large classes of finite dimensional representations for both. In this talk, I will discuss examples of those representations and their properties. An important feature is that they are local under cutting and gluing of handlebodies. I will sketch an approach to a precise formulation of this property and present a complete classification result. The talk is based on joint work with Lukas Woike.

Colloquium

Speaker: Craig Kaplan (Waterloo)

"Aperiodic Monotiles"

Time: 15:30

Room: MC 107

A set of shapes is called aperiodic if they admit tilings of the plane, but none that have translational symmetry. Starting in the 1960s, progress in our understanding of such tilings produced progressively smaller aperiodic sets, but failed to arrive at a single shape that tiles aperiodically, also known as an aperiodic monotile. In 2023, we resolved that open question by proving that the "hat", a union of eight kites, is an aperiodic monotile. In this talk I provide background on aperiodicity and related topics in tiling theory, review the history of the search for for an aperiodic monotile, and present the new monotiles that we discovered.

Graduate Seminar

Speaker: Alejandro Santacruz Hidalgo (Western)

"Hardy's inequality: a brief review, some extensions and applications"

Time: 15:30

Room: WSC 187

In 1915 G.H Hardy needed an estimate for arithmetic means to find a proof of Hilbert's inequality for sequences, a continuous version of that inequality followed in 1925. Since then, extensions have been made in many directions; more general domains, weighted norm inequalities, general measures, among others. In this talk we will review the classic statement of Hardy's original inequality. We will explore some of the extensions of this important inequality and review some of its implications, such as, Sobolev inequalities and boundedness of the Fourier transform in weighted Lorentz spaces.

Transformation Groups Seminar

Speaker: Vladimir Gorchakov (Western)

"3-dimensional small covers and links"

Time: 10:30

Room: MC 108

Let M be an orientable 3-dimensional small cover, i.e. a real analog of quasi-toric space with an action (Z_2)^3. Let G be a subgroup of orientation-preserving homeomorphisms in (Z_2)^3. In this talk we will discuss an orbit space of M/g for g in G. We will show that M/g is homeomorphic to S^3 or connected sum of several copies of S^1 x S^2. In the case when the orbit space is S^3 we will describe M as the double-branched covering of S^3 over a link and relate it to Hamilton cycles in polytopes. We also will discuss how this result is related to rigidity problems in toric topology.

Analysis Seminar

Speaker: Rasul Shafikov (Western)

"Rational convexity and symplectic geometry, II"

Time: 14:30

Room: MC 108

In this talk I will define Liouville vector fields, Weinstein domains, contact type submanifolds and discuss their connection with Stein structures and rational convexity in $\mathbb C^n$.

Geometry and Topology

Speaker: Dan Christensen (Western)

"H-spaces and infinite loop spaces"

Time: 15:30

Room: MC 107

An H-space is a topological space with a unital binary operation. The theory of H-spaces goes back to the 1950's and is closely connected to the theory of loop spaces. In this talk, I'll give some background about H-spaces and then present several new results in this area, including a property that allows us to deloop certain "central" H-spaces. While this work was done in the framework of homotopy type theory, and therefore applies in any $\infty$-topos, I will explain it in the special case of topological spaces.

This is based on joint work with Ulrik Buchholtz, Jarl Flaten, and Egbert Rijke in arxiv:2301.02636.

Colloquium

Speaker: (Western)

"Brauer invariants of finite groups"

Time: 11:00

Room: MC 108

Algebra Seminar

Speaker: Stefan Gille (University of Alberta)

"TBA"

Time: 14:30

Room: MC 108

TBA

Transformation Groups Seminar

Speaker: Li Cai (Xi'an Jiaotong-Liverpool University)

"On Graph Products of Simplicial Groups"

Time: 09:00

Room: Zoom

In this talk we give a homotopical interpretation of a result of Panov and Veryovkin on the commutator subgroup of a graph product of discrete groups, using the language of simplicial groups. It turns out that the commutator subgroup of a graph product of simplicial groups gives a model of the loop spaces of certain polyhedral products. The Pontryagin algebra of these loop spaces will also be discussed. We will also introduce some questions that require further work.

Geometry and Topology

Speaker: Taylor Brysiewicz (Western)

"Algebraic Matroids, Monodromy, and the Heron Variety"

Time: 15:30

Room: MC 107

Heron's formula gives the area of a triangle in terms of the lengths of its sides. More generally, the volume of any simplex is determined by its edge-lengths via a Cayley-Menger determinant.

In work-in-progress with Seth Asante and Michelle Hatzel, we ask

*Which other sets of volumes of faces of an n-simplex, when known, determine the remaining unknown face-volumes?*.

An answer to this question is encoded in the algebraic matroid of the Heron variety. Moreover, we ask

*When are these unknown volumes recoverable via formulae in the known volumes?*

We answer these questions for n<5 by combining techniques in computational group theory, computer algebra, field theory, and numerical algebraic geometry. Of particular focus is the problem of recovering the 10 edge lengths of a 4-simplex from its 10 triangular face areas, a problem motivated by applications in theoretical physics.

Ph.D. Public Lecture

Speaker: Jacob Imre (Western)

"Asymptotic Approximations of Lambert W and Related Functions"

Time: 10:00

Room: MC 108

In the realm of multivalued functions, certain specimens run the risk of being elementary or complex to a fault. The Lambert W function serves as a middle ground in a way, being non-representable by elementary functions yet admitting several properties which have allowed for copious research. W utilizes the inverse of the elementary function xe^x , resulting in a multivalued function with non-elementary connections between its branches. W_k(z), the solution to the equation z = W_k(z)e^W_k(z) for a “branch number” k ∈ Z, has both asymptotic and Taylor series for its various branches.

This lecture will primarily focus on the asymptotic series approximations of any branch of the lambert W function. The series previously used to approximate the "principal" branch of the Lambert W can be generalized to any branch. Similarly, the presentation will focus on improvements to these series and why they're possible. Also included is a section on a family of functions closely related to Lambert W, and how their asymptotic approximations can be derived from W's.

Colloquium

Speaker: Richard Szabo (Edinburgh)

"Random partitions, instantons and enumerative geometry"

Time: 15:30

Room: MC 107

Counting partitions in diverse dimensions is a long-standing problem in enumerative combinatorics. It also plays a prominent role in the physics of instanton counting and in algebraic geometry through the computation of Donaldson-Thomas invariants. In this talk I will give an overview of these counting problems, and discuss how recent developments in the computation of instanton/Donaldson-Thomas partition functions clarify some open problems in the enumeration of higher-dimensional partitions.

Algebra Seminar

Speaker: Aaron Landesman (MIT)

"The distribution of Selmer groups and ranks of abelian varieties in quadratic twist families over function fields"

Time: 14:30

Room: MC 108

The minimalist conjecture predicts that, in quadratic twist families of abelian varieties, half have rank 0 and half have rank 1. This fits into the larger picture of the Bhargava-Kane-Lenstra-Poonen-Rains heuristics, which predict the distribution of Selmer groups of these abelian varieties. In joint work with Jordan Ellenberg, we prove a version of these heuristics: over function fields over the finite field $\mathbb{F_q}$, we show that the above heuristics are correct to within an error term in $q$, which goes to 0 as $q$ grows. The main inputs are a new homological stability theorem in topology for a generalized version of Hurwitz spaces and an expression of average sizes of Selmer groups in terms of the number of rational points on these Hurwitz spaces over finite fields.

Graduate Seminar

Speaker: Elaine Murphy (Western)

"The Mathematical Structure of Point Mutations"

Time: 15:30

Room: MC 107

Mutation is the engine of evolution. By considering only single point mutations (SNPs) on DNA sequences, we see a natural group theoretic model of mutations acting on the set of nucleotides. In this talk, we will investigate the implications of this structure for synonymous mutations (mutations that do not change the encoded amino acids) and how this affects the notion of distance between two genetic sequences.

Analysis Seminar

Speaker: Michael Francis (Western)

"The b-Newlander-Nirenberg theorem"

Time: 14:30

Room: MC108

Melrose introduced the formalism of b-geometry as a tool for studying partial differential operators on a smooth manifold M that suffer a first order degeneracy along a given hypersurface Z. The b-tangent bundle is the vector bundle whose sections are smooth vector fields defined on all of M and tangent along Z. Many of the classical geometries admit "b-analogues" in which the b-tangent bundle fills the role of the usual tangent bundle (so one has symplectic b-geometry, Riemannian b-geometry, etc). Complex b-geometry was introduced by Mendoza. In this talk, we will discuss the "b-Newlander-Nirenberg theorem": every complex b-manifold is locally isomorphic to some standard model. This allows one to define complex b-manifolds in the way one might hope, in terms of appropriately defined "b-holomorphic charts". This is joint work with Tatyana Barron.

Geometry and Topology

Speaker: Matthias Franz (Western)

"An $A_\infty$-version of the Eilenberg-Moore theorem"

Time: 15:30

Room: MC 107

I will discuss a new product structure on the two-sided bar constructions of singular cochains. This bar construction is used in the Eilenberg-Moore theorem to compute the cohomology of pull-backs of fibrations. The product structure is based on so-called homotopy Gerstenhaber operations on singular cochains, transforming the two-sided bar construction into an $A_\infty$-algebra. This type of algebra is associative up to a strong form of homotopy. The new product includes those previously defined by Baues, Gerstenhaber-Voronov, Kadeishvili-Saneblidze, and Carlson-Franz as special cases. Consequently, the multiplicative cohomology isomorphism from the Eilenberg-Moore theorem is elevated to a quasi-isomorphism of $A_\infty$-algebras.

Also, please make sure the event does not appear twice on the calendar. Thanks!

Colloquium

Speaker: Lyle Muller (Math department, Western)

"Spatiotemporal dynamics in neural systems: from data to mathematical models and computation"

Time: 15:30

Room: MC 107

Neurons in cortex are connected in intricate patterns, with local- and long-range connections and time delays for transmitting signals. In recent work, we have found that spontaneous and stimulus-driven waves travel over these networks, changing excitability of the neurons and shaping perceptual sensitivity. Understanding how these networks generate these sophisticated dynamics, however, remains an open problem. This is due, in part, to the fact that connecting the specific structure of networks to the nonlinear dynamics that will result is a difficult problem in general. Further, experiments suggest one mechanism for these waves could be the distance-dependent time delays due to transmitting spikes along the axons connecting neurons across these networks. Analyzing the underlying network mechanism for these waves thus represents an additional challenge, as we need to consider systems with many time delays.

In this talk, I will present recent results from my research team connecting the structure of individual networks to the resulting dynamics in systems of nonlinear Kuramoto oscillators. We introduce a complex-valued approach to the Kuramoto model that allows connecting the eigenspectrum of the graph adjacency matrix to the nonlinear dynamics that result in individual simulations of this system. This approach allows predicting the specific spatiotemporal pattern that will result from the connectivity pattern in an individual network. An extension of this approach allows predicting the specific spatiotemporal patterns generated by distance-dependent time delays from spike transmission in these systems. Finally, I will present our latest efforts to understand computation with spatiotemporal dynamics in neural systems using these nonlinear network models.

Graduate Seminar

Speaker: Prakash Singh (Western)

"The Hofer diameter problem for rational symplectic manifolds"

Time: 15:30

Room: MC 107

In general, Lie groups do not admit bi-invariant metrics, and infinite dimensional Lie groups should not admit such metrics either. But surprisingly, Ham admits one such metric (in fact, unique in a sense), called the 'Hofer metric', discovered by Hofer in the 90s. People have been studying the large-scale geometry properties of this metric for a long time, but such studies were restricted to either 2-dimensions, monotone symplectic manifolds, or to aspherical manifolds. In particular, it is widely conjectured that the hofer diameter is infinite for every closed symplectic manifold, and this conjecture has been settled for the above-mentioned manifolds. I will talk about the diameter problem associated with this metric for some rational ruled manifolds like CP2, S2 x S2, and their blow-ups, using methods from quantum homology and spectral invariants on them. I will prove the conjecture for CP2 and S2 x S2, and I will prove it under a mild assumption (but unproven) for S2 x S2 blown up once.